 All right, welcome back today in this video. We're going to talk about solving a quadratic equation by completing the square Now I do have a I do have a excuse me I got a little little something in my throat here if you have if you don't know what completing the square is I do have a previous video that I did about completing the square Please go watch that first or have a good understanding of how to complete the square before you watch this or else You're going to be pretty lost on what we're doing here Okay, so solving a quadratic equation by completing the square here We go the first thing you want to do is you want to set everything up Okay, now this isn't this isn't a normal quadratic equation This isn't a normal way to solve it what you need to do is basically what's why I tell students Take your x's and put them on one side of your equal sign take your numbers and put them on the other side So here's my equal sign put the numbers over here equal sign Equal sign put the variables over here equal sign put the numbers over here And then also when you're setting this up add some blank spots those blank spots are going to be used for when we actually Complete the square, okay, and I'll explain why we have two of them here in just a minute Okay, now this over here when if you watch my previous video this right here should be should be familiar We have this x squared minus 12x plus some number We need to figure out what this number is so we can factor here in just a moment Okay, so that number how do we find it? We take b divided by 2 and we square it Okay, so in this case our b number is negative 12 negative 12 divided by 2 and then square it negative 12 divided by 2 is negative 6 Negative 6 squared is going to be a 36 So I have a 36 that I got to add to this left side over here now Going way back to the your previous days of pre-algebra on that kind of stuff when you first learn how to solve equations Whatever you do to one side of your equation You must also do to the other so if I got to add 36 to one side here I also got to add 36 to the other side got to keep the equation balanced must keep the equation balanced All right, so now what I can do is actually on this right side negative 20 and a positive 36 That just gives me 16 that actually simplifies really nicely on this other side here This is where I'm going to factor. This is where I'm going to do my parentheses I got x's in the beginning bubbles here numbers that multiply to 36 Positive 36 and then add a negative 12 our negative 6 and negative 6 and again If you do you're completing the square process, right? If you do this b divided by 2 squared if you do that, right and you do your factoring, right? You're always get the same parentheses and that's kind of the idea with Completing the square is you're always going to get the same parentheses it makes the factoring so easy here We go now after this condense this down To the same parentheses condense that to just a single parentheses squared So this is x minus 6 quantity squared Equals 16. Okay. Now if I want to solve for x. I'm solving this equation solving for x I still need to get x by itself, right? So I need to get rid of this squared and rid of this negative 6 so to get rid of squaring I need to square root okay squaring and square rooting are opposite operations They cancel each other out, but if I square root one side, I must also square root the other side now Here's the here's the trick though this little symbol if you if you Introduce the square root symbol to the equation that means that you need to also think about the positive square root and the negative square root that you could get from square rooting a number because a Positive 4 times 4 gets you back to 16 But a negative 4 and a negative 4 also multiply back to get you 16 So that's that that's kind of a short version of why a Short easy version of understanding why we have to do this little positive negative symbol here, okay? These cancel I'm left with x minus 6 there on the left side Okay, now to continue to solve take the 6 add it over to the other side x equals 6 plus or minus 4 x Plus or minus 4 okay now what this does is this get this is actually going to give us two answers Okay, I have two operations here. I have a plus and a minus so this is going to be 6 plus 4 and 6 minus 4 x equals 6 plus 4 is 10 and Then 6 minus 4 is going to be 2 I have two solutions to this equation You're gonna get that most of the time not all the time, but most of the time you're gonna get two solutions to any quadratic equation now if you think about it just a little bit a Quadratic equation looks like this you've heard to graph it looks like a parabola Okay, it looks like a parabola this parabola would look something like this Two here ten here My parabola would look something like this Will you ship there with a two here and a ten here? Okay, that gives you kind of an idea of when you solve a quadratic equation That if you were to graph it, this is kind of what it would look like Okay, so I kind gives you an idea of why we solve it and all that kind of stuff what we can find from it Now if this graph doesn't convince you of your answer You can also take these numbers and just plug them back up into your original answer your original problem And it will give you the answer I mean ten squared is is a hundred and then twelve times ten is 120 120 minus 20 is gonna give you hundreds You got a hundred on the left hundred on the right when you do two you get the same thing You I think you get four on the left and four on the right But anyway now I'm gonna go over to this next example over here. I'm gonna change my colors just a little bit So we don't get confused Mix up the colorings here. Okay, this this equation a little bit more complicated It looks a little bit more complicated But it just basically what it means is you just have a couple of steps You got to do before you start completing the square notice here everything's out of order this thing is a mess Everything has out of order. Okay, you're x squared. They're supposed to go first. This needs to be in front Three x squared plus 18 x equals 45. That's what it's supposed to look like Okay, and get your get your x's on one side your numbers on the other Okay, now what I'm gonna do though is and notice over here we need to have a one in front of x squared So this three that's gonna cause some troubles for us We cannot complete the square with a number other than one in front of there So that three has got to go but that basically means is though I just need to divide everything by three if you divide by three it'll make a one here But if you divide one of your terms for your equation by three you gotta divide everything by three But notice that well everything is divisible by three so it's kind of set up that way Now what I'm gonna do is I'm gonna do a couple of things here. That's not 18 Supposed to be six six x I want to divide all the numbers But then I'm also gonna leave spaces for what I'm gonna complete my square Okay, so there we go Three divide by three is one 18 divided by three is six and 45 divide by three is 15 So there we go. So now what I'm going to do is complete my square Okay, I'm going to find the number that I need that goes in this spot So I take my I take that B number which in this case is six if you can read that six Is my B number divide by two and squared so six divide by two is three Three squared is nine. So nine is the number that I'm going to add to both sides. Okay now on this right side You can simply just add this together which gives you a 24 now if you're foreshadowing a little bit over here We had 16 which reduced really very nicely to plus or minus four over here It's not going to be we're not going to have that that luxury but we'll get that here in a minute this year. I'm going to factor In this case, I get x plus three x plus three Okay, now I'm going to condense that down just like the last one x plus three quantity squared equals 24 Now I'm going to square root both sides get rid of the squared x plus three quantity squared Try not to go too fast through this plus minus the square root of 24 Okay, now got the plus minus symbol in there square root of 24 now again. I don't know what the square root of 24 is It's if you think it's 4.9. It's blah blah blah blah something like that but in this case We're not going to use decimals. We're not going to use messy decimals. I hate messy decimals They just don't they don't work out for anybody. Okay, so just leave it as a square root of 24 for right this second We're actually going to reduce that But let me get this left side here Up a little bit. So we got x plus three on this left side this right side is gonna be plus minus now as I said before That 24 I'm going to reduce that just a little bit now. Remember when you reduce radicals split it up Using a perfect square. So four is a perfect square four times six gives you back to 24 Now the reason we use a perfect square The reason that we use a perfect square is because we actually know what the square root of a perfect square is so in this case The square root of four is to 16 over here That is a perfect square because we know the square root of it is four Four is a perfect square because we know the square root of it is two six is not a perfect square because we don't know What the square root of six is so we just kind of leave it to root six messy decimals 4.9 blah blah blah something like that. Just leave it as to root six. Okay, depending on the type of problems that you have Depending on the type of problems that you have sometimes you use decimals. Sometimes you won't but in this case I am suggesting that you do not So I take this positive three subtract it over to the other side. Do not do not mix those up Do not make messy decimals? Hate messy decimals, but anyway, it also depends on what your teacher wants Make sure you ask your teacher. Do you want this is what's called exact form? Exact form Okay, that's exactly what the answer is. It's nice. It's it's nice numbers It's easy to read that kind of stuff if you actually punch that into the calculator If you take negative three plus two root six and negative three minus two root six You get just some messy decimals. I think it's negative one I can't remember exactly what it is But again, I don't I don't like it. I don't want to remember what it is. It's messy decimals Don't want them. Okay, but make sure you ask your teacher when you're going through this Do you want debt you want exact form or do you want decimals rounded to the? Thousandth place or to five decimal places or what do you want to make sure you ask? Okay, so and now what I'm going to do is very quickly very very quickly go over a a way of us Using the quadratic formula for something okay for something other than just solving equations. So here we go Write the following function in vertex form and identify its vertex now. Notice that this this equation is in standard form Okay, x squared sixteen x negative twelve get your squareds x's and constants everything is in standard form Okay, what I'm going to do is I'm going to change this I'm going to use my completing the square process to change this into vertex form the vertex form of a Quadratic equation tells you where the vertex of the parabola is going to be this vertex as many many uses it tells you Maximums minimums it tells you the maximum height or the lowest height whatever it is So if you if you're using it for a real-life problem, it just it tells you so many different things Okay, but again depending on those examples I don't have a good one. Well, I guess if you shoot a basketball if you shoot a basketball That's a parabolic function Okay, because the ball going to go ball leaves your hand it goes up and then it comes back down then it goes into the basket Okay, that that right there. There's that's a parabolic path. That's a parabola. Okay, so the vertex that that's the highest That basketball is going to go you can actually figure that out. Okay now in this case The f of x just leave it there. Just leave this function notation there x squared plus sixteen x plus something So right there there's something familiar. That's our That's where we're going to complete the square minus 12 minus blank now I'll explain why I put a minus blank there here in a minute because it looks very different from the previous problem Okay, just just give me a moment All right, so now what I'm going to do is figure out how to complete the square So take that 16 number take that B number Divide by 2 and square it so 16 divided by 2 is 8 8 squared is 64 So it's 64 that I'm going to add here, but now here's the trouble you can't just add 64 to one side willy-nilly just whatever you want to you can't do that There are rules you have to follow if I add something to one side I have to add it to the other now for the sake of simplicity. I did not put my blank spot over here I left it over here. So but the thing is is I'm gonna put my 64 here. It's a negative 64 It's a minus 64, which I did on purpose If you're if you're ever gonna do this technically what I'm doing is I'm adding 64 to here And I'm subtracting 64 over here Technically, I'm adding nothing to one side plus 64 and a minus 64 Technically, I'm adding nothing to this one side. So it's it's okay I'm keeping my equation balanced, which which is the general rule that you gotta follow keep the equation balanced Okay, now what I'm going to do is I'm going to take this section here Okay, be careful wall it off from everything else take this section here and factor it x plus 8 x plus 8 Minus 12 and minus 64 these can combine to a negative 76 Okay, now some of you who know what vertex form is you're gonna start seeing it already Okay, this is gonna be x plus 8 quantity squared minus 76 this right here. This is vertex form We're halfway done with this problem write the following function in vertex form and identify its vertex So now I got to identify the vertex So now the vertex is x y coordinates now, which is really nice because here's your x y coordinates right here That's why we call it vertex form. It shows us what the vertex is now This tells you the y coordinate because this tells you if you studied transformations of quadratic functions This right here tells you how to go down 76 Okay, so I actually tells you the y coordinate this one over here This is kind of opposite of what you think it's right next to the x so we know it's it's horizontal It's gonna move left and right, but it's opposite of what you think it's not gonna be to the right eight This one is gonna be to the left eight, so it's actually gonna be a negative eight is where our vertex is going to be So that's a little bit different there. Okay, so our function is x plus 8 quantity squared minus 76 and our vertex is it gonna be at negative 8 negative 76 Okay, I do go through that a little quickly. I want to try to make this video short, but anyway I Hope you enjoyed the video. I know I thoroughly enjoyed making these videos for you for you guys But I hope you enjoyed the video. I hope you guys learned something today, and we'll see you next time